Flows and Networks Plan for today (lecture 4):

1. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

2. Poisson process • Definition : Poisson process :Let S1,S2,… be a sequence of independent exponential() r.v. Let Tn=S1+…+Sn, T0=0 and N(s)=max{n,Tn≤s}. The counting process {N(s),s≥0} is called Poisson process. • Theorem : If {N(s),s≥0} is a Poisson process, then(i) N(0)=0,(ii) N(t+s)-N(s)=Poisson( t), and(iii) N(t) has independent increments.Conversely, if (i), (ii), (iii) hold, then {N(s),s≥0} is a Poisson process • PASTA

3. Let t0 fixed. Arrival process Poisson, thus arrival process after t0 independent of number in queue at t0. For reversed process X(-t): arrival process after –t0 independent of number in queue at –t0 Reversibility: joint distribution departure process up to t0 and number in queue at t0 for X(t) have same distribution as arrival process to X(-t) after –t0 and number in queue at –t0. In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0 Holds for each reversible Markov process with Poisson arrivals as long as an arrival causes the process to change state Output simple queue

4. Simple queue, Poisson() arrivals, exponential() service Equilibrium distribution Tandem of J M/M/1 queues, exp(i) service queue i Xi(t) number in queue i at time t Queue 1 in isolation: simple queue. Departure process queue 1 Poisson, thus queue 2 in isolation: simple queue State X1(t0) independent departure process prior to t0,but this determines (X2(t0),…, XJ(t0)), hence X1(t0) independent (X2(t0),…, XJ(t0)). Similar Xj(t0) independent (Xj+1(t0),…, XJ(t0)). Thus X1(t0), X2(t0),…, XJ(t0) mutually independent, and Tandem network of simple queues

5. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

6. Waiting time simple queue (1) • Consider simple queue FCFS discipline • W : waiting time typical customer in M/M/1(excludes service time) • N customers present upon arrival • Sr (residual) service time of customers present PASTA Voor j=0,1,2,…

7. Waiting time simple queue (2) • Thus • is exponential (-)

8. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

9. Sojourn time tandem simple queues Recall: In equilibrium the departure process from an M/M/1 queue is a Poisson process, and the number in the queue at time t0 is independent of the departure process prior to t0 Theorem 2.2: If service discipline at each queue in tandem of J simple queues is FCFS, then in equilibrium the waiting times of a customer at each of the J queues are independent Proof: Kelly p. 38 Tandem M/M/s queues: overtaking Distribution sojourn time: Ex 2.2.2

10. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

11. Little’s law (1) • Let • A(t) : number of arrivals entering in (0,t] • D(t) : number of departure from system (0,t] • X(t) : number of jobs in system at time t Equilibrium for t∞ In equilibrium: average number of arrivals per time unit = average number of departures per time unit

12. Little’s law (2) Fj sojourn time j-th departing job S(t) obtained sojourn times jobs in system at t

13. Little’s law (3) Assume following limits exist(ergodic theory, see SMOR) Then Little’s law

14. Little’s law (4) • Intuition • Suppose each job pays 1 euro per time unit in system • Count at arrival epoch of jobs: job pays at arrival for entire duration in system, i.e., pays EF • Total average amount paid per time unit  EF • Count as cumulative over time: system receives on average per time unit amount equal to average number in system • Amount received per time unit EX • Little’s law valid for general systems irrespective of order of service, service time distribution, arrival process, …

15. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

16. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

17. Partial balance equations • The boundary ni=0 • Decomposition of global balance • Interpretation

18. Flows and NetworksPlan for today (lecture 4): • Last time / Questions? • Waiting time simple queue • Sojourn time tandem network • Little’s law • Jackson network: mean sojourn time • Partial balance revisited • Blocking of transitions • Summary / Next • Exercises

19. Simple queues, exponential service queue j, j=1,…,J statemovedepartarrive Transition rates Traffic equations Solution Blocking in tandem networks of simple queues (1)

20. Simple queues, exponential service queue j, j=1,…,J Transition rates Traffic equations Solution Equilibrium distribution Partial balance PICTURE J=2 Blocking in tandem networks of simple queues (2)

21. Simple queues, exponential service queue j, j=1,…,J Suppose queue 2 has capacity constraint: n2<N2 Transition rates Partial balance? PICTURE J=2 Stop protocol, repeat protocol, jump-over protocol Blocking in tandem networks of simple queues (3)

22. Summary / next: Equilibrium distributions • Reversibility • Output reversible Markov process • Tandem network • Jackson network • Partial balance • Kelly-Whittle network NEXT: Optimization (W sec 9.7) Quasi-reversibility (R+SN Ch 3)

23. Exercises • Exercise BlockingConsider a tandem network of two simple queues. Let the arrival rate to queue 1 be Poisson , and let the service rate at each queue be exponential i , i=1,2. Let queue 1 have capacity N1. Queue 2 is a standard simple queue. For N1= , give the equilibrium distribution. For N1<  formulate three distinct blocking protocols that preserve product form, indicate graphically what the implication of these protocols is on the transition diagram, and proof (by partial balance) that the equilbrium distribution is of product form.

24. Exercises [R+SN] 2.1.1, 2.1.2, 2.3.1, 2.3.4, 2.3.5, 2.3.6, 2.4.1, 2.4.2, 2.4.6, 2.4.7